Understanding the equation of TD(0) in the paper "Learning to predict by the methods of temporal differences"
When lambda = 0 as in TD(0), how does the method learn? As it appears, with lambda = 0, there will never be a change in weight and hence no learning. 


I think the detail that you're missing is that one of the terms in the sum (the final "iteration" of the sum, the case where $k = t$) has $\lambda$ raised to the power $0$, and anything raised to the power $0$ (even $0$) is equal to $1$. So, for $\lambda = 0$, your update equation becomes

$$\Delta w_t = \alpha \left( P_{t+1} - P_t \right) \nabla_w P_t,$$

which is basically a one-step update (just like Sarsa).